Spectral Theory for Borel PMP Graphs

TL;DR

This paper develops spectral theory for bounded-degree Borel pmp graphs, establishing new bounds on chromatic numbers, a measurable Tutte condition, and spectral continuity results.

Contribution

It introduces a systematic spectral framework for Borel pmp graphs, including bounds on chromatic numbers and a measurable Tutte condition, advancing the understanding of their spectral properties.

Findings

Spectral characterization of approximate measurable bipartiteness

Upper bounds on approximate measurable chromatic number

Spectral continuity under local-global convergence

Abstract

We initiate a systematic study of spectral theory for bounded-degree Borel pmp graphs. Specifically, we study spectral properties of the associated adjacency and Laplacian operators. We start with proving a spectral characterization of approximate measurable bipartiteness. Next, we adapt classical theorems of Wilf and Hoffman to give novel upper and lower bounds on the approximate measurable chromatic number. Using similar techniques, we then show that the approximate measurable chromatic number of a pmp graph generated by $n$ bounded-to-one functions is at most $2n + 1$. Next, concerning matchings, we introduce a measurable version of Tutte's condition and show that a spectral assumption analogous to the one from a classical theorem of Brouwer and Haemers implies this measurable Tutte condition. Finally, we show that the spectrum is continuous under local-global convergence.